Problem 31 - Precalculus: A Concise Course

Question

Graph \(f\) and \(g\) on the same set of coordinate axes. (Include two full periods.) $$ \begin{array}{l} f(x)=-2 \sin x \\ g(x)=4 \sin x \end{array} $$

Step-by-Step Solution

Answer

The function \(f(x)\) has an amplitude of 2 and is reflected in the x-axis. The function \(g(x)\) has an amplitude of 4. Both functions have a period of \(2\pi\).

Step 1: Identify characteristics of each function

For the function \(f(x)=-2 \sin x\), it has an amplitude of 2 and is a reflection in the x-axis of the standard sine function. For the function \(g(x)=4 \sin x\), it has an amplitude of 4. Both functions have a period of \(2\pi\) because the coefficient of \(x\) in both functions is 1.

Step 2: Plot the functions

Begin by sketching the x and y axes. Mark off a distance of \(2\pi\) along the x-axis to represent one period for both functions. Then, divide this interval into four equal parts to mark the key points of the sine functions (0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\)). For \(f(x)\), put the peaks at -2 and the troughs at 2. For \(g(x)\), put the peaks at 4 and troughs at -4. Join these points using the characteristic 'S' shape of the sine function to create the graphs.

Key Concepts

Amplitude of Sine Wave

Understanding the amplitude of a sine wave is crucial when graphing sine functions like those in the given exercise. The amplitude represents the wave's maximum distance from its central axis (usually the x-axis in a graph). It indicates how high or low the wave peaks and troughs go, providing a measure of the wave's strength or intensity.

For the function f(x) = -2 sin x, the amplitude is found by taking the absolute value of the coefficient before the sine function, which in this case is 2. Despite the negative sign, which indicates a reflection across the x-axis, amplitude is always a positive value, signifying that the wave peaks at -2 and troughs at +2 for this function.

Similarly, for g(x) = 4 sin x, the amplitude is 4, meaning this wave reaches higher peaks and lower troughs than f(x), with its maximum and minimum values being 4 and -4, respectively. So, when graphing, the waves of f(x) and g(x) will fluctuate with these amplitudes for two full periods.

Period of Sine Function

The period of a sine function is the horizontal length it takes for the wave pattern to repeat itself. In more technical terms, it's the smallest interval x after which the function's values start repeating. For basic sine functions like those in our exercise, the period is typically 2π, since the sine function naturally repeats every 2π radians.

In both f(x) and g(x), the coefficient of x inside the sine function is 1. This coefficient (often denoted as B in sin(Bx)) determines the period of the function. The period P can be calculated using the formula P = 2π/|B|. In our functions, since B is 1, the period remains 2π. This means that the graphs of f(x) and g(x) complete one full cycle of rising to a peak, descending to a trough, and returning to the starting level every 2π units along the x-axis.

The step-by-step solution indicates this by dividing the x-axis into intervals of π/2 to map the key points of one period (0, π/2, π, 3π/2, 2π). Understanding this concept is fundamental when predicting the wave's behavior at different points along the x-axis.

Sine Function Transformations

Graphing sine function transformations involves applying various changes to the base sine function, such as stretching, compressing, reflecting, translating, or any combination of these actions. The general form of a transformed sine function is y = A sin(B(x - C)) + D, where A, B, C, and D affect different aspects of the graph.

A affects the amplitude as discussed earlier, B influences the period, C corresponds to horizontal shifts, and D represents vertical shifts. For the given functions, f(x) and g(x), the only transformations from the base sin x function are reflections and amplitude changes—there are no shifts or period changes because C and D are zero, while B is 1.

The negative sign in f(x) is a reflection transformation that flips the graph across the x-axis. Therefore, while the amplitude stays at 2, the peaks and troughs are inverted for f(x) compared to the standard sin x curve. There are no horizontal or vertical shifts for either function as the center of each wave lies on the x-axis, and they pass through the origin. Correctly applying these transformations is essential for accurately sketching complex sine graphs beyond the textbook basics.